The orbit of the circumbinary planet Kepler-16b is significantly non-Keplerian because of the large secondary-to-primary mass ratio (0.29) and orbital eccentricity (0.15) of the binary, as well as the proximity of the planet to the binary (orbital period ratio 5.6). We present an analytic theory which models the motion of the planet (treated as a test particle) by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the binary's potential, the epicyclic motion, and the vertical motion. In this analytic theory, the periapse and ascending node of the planet precess at nearly equal rates in opposite directions, and the largest forced oscillation term corresponds to a forced eccentricity of 0.035. The nodal precession period (42 years) found in direct numerical orbit integration is in excellent agreement with the analytic theory, while the periapse precession period (49 years) and forced eccentricity (0.038) are slightly larger than the analytic values. The comparison with direct numerical orbit integration also shows that the planet's orbit has a nonzero epicyclic (or free) eccentricity of 0.027.

The orbit of the circumbinary planet Kepler-16b is significantly non-Keplerian because of the large secondary-to-primary mass ratio (0.29) and orbital eccentricity (0.15) of the binary, as well as the proximity of the planet to the binary (orbital period ratio 5.6). We present an analytic theory which models the motion of the planet (treated as a test particle) by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the binary's potential, the epicyclic motion, and the vertical motion. In this analytic theory, the periapse and ascending node of the planet precess at nearly equal rates in opposite directions, and the largest forced oscillation term corresponds to a forced eccentricity of 0.035. The nodal precession period (42 years) found in direct numerical orbit integration is in excellent agreement with the analytic theory, while the periapse precession period (49 years) and forced eccentricity (0.038) are slightly larger than the analytic values. The comparison with direct numerical orbit integration also shows that the planet's orbit has a nonzero epicyclic (or free) eccentricity of 0.027.